Stongly Coupled Field Theories, String Theory and Gravity

本项目涉及弦理论和量子场论（QFT）。有两大目标。第一部分是使用受弦理论启发的工具来描述难以处理的QFT体系。第二部分是使用弦理论中的新几何工具来描述引力和我们可观测宇宙的各个方面。QFT非常好地描述了我们宇宙亚原子水平上的相互作用，支撑了所有实验验证的粒子和相互作用。除特殊情况外，量子傅立叶变换的方程是笨拙的，不适于直接分析。标准的方法是用一种称为摄动理论的方法来近似这些方程。这要求粒子之间的相互作用是弱的，而不是普遍的情况。通常，相互作用是强的，一种称为强耦合的情况，并且微扰理论近似失效。这个问题是理解粒子物理学许多方面的主要限制。此外，量子傅立叶变换并不描述宏观相互作用，特别是引力。如果我们处理的是非常密集的物体，比如黑洞，那么我们需要找到一种结合爱因斯坦广义相对论和量子力学理论的理论。这方面的主要候选理论是弦理论。为了工作，它需要施加严格的数学条件。例如，除了我们所观察到的三维空间之外，一定还存在着六个额外的空间，它们的几何形状非常小，所以在今天的实验中是看不见的。一个粗略的类比是软管：从远处看，它看起来是一维的，但仔细观察，它有一个额外的圆形方向。描述可观测宇宙的物理变成了一个与特定空间的几何密切相关的问题，反过来，要求合理的物理作为弦理论的输出导致新的几何技术。我们可以把量子修正理解为来自弦理论本身。弦理论为我们理解量子傅立叶变换和引力带来了新的思路。我们从强耦合QFT开始。全息是两种理论之间的等价，我们称它们为理论A和理论B。理论A是d+1维引力理论，而理论B是平坦（无重力）d维空间中的量子ft。全息意味着理论A可以用来学习理论B的强耦合方面，反之亦然。例如，理论B可以是可积的，这意味着它是完全可溶的，关于强耦合的信息完全由对称性决定。全息技术意味着我们可以确定（也许是模糊的）引力理论A的特性。相反，通过经典引力，理论A可以用来描述理论B中的强耦合状态。即使理论A和理论B都不是现实模型——人们通常会对对偶性进行简化假设——人们可能希望得到的特征是普遍的，在粒子物理中其他困难的问题上给我们一些新的教训。本提案的第一部分是关于在一个新的范例中发展全息，以及使用可积性来探索量子傅立叶变换的性质。接下来，在引力的背景下，在理解黑洞方面出现了新的想法。来源于弦理论的对称性，例如超对称性，已经导致了获得新型黑洞的新技术，以及理解它们的几何和物理特性。许多有趣的问题出现了：量子修正对这些黑洞解的作用是什么？它们稳定吗?另一个不同但相关的问题是我们如何用弦理论来描述准现实现象学模型？这样做需要理解空间的几何结构。什么样的几何图形能得出我们宇宙的现实模型？量子修正的作用是什么？这些是构成本建议第二部分的问题类型。

This project is concerned with string theory and quantum field theory (QFT). There are two broad aims. Part I is to use tools inspired from string theory to describe otherwise intractable regimes of QFT. Part II is to use new geometric tools within string theory to describe aspects of gravity and our observable universe. QFT describes interactions at the sub-atomic level of our universe exceptionally well, underpinning all experimentally verified particles and interactions. The equations of QFT, except in special circumstances are unwieldy and not amenable to a direct analysis. The standard approach is to approximate the equations in a manner known as perturbation theory. This requires the interactions between particles be weak, not a universal situation. Often, the interactions are strong, a situation known as strong coupling, and the perturbation theory approximation breaks down. This problem is a major limitation for understanding many aspects of particle physics. Moreover, QFT does not describe macroscopic interactions, in particular gravity. If we are dealing with very dense objects such as black holes, then we need to find a theory that incorporates Einstein's theory of general relativity and QFT. The leading candidate that does this is string theory. In order to work, it requires stringent mathematical conditions be imposed. For example, in addition to the three dimensions we observe, there must exist six additional dimensions, whose geometry is very small and so not visible to present day experiment. A rough analogy is with a hose: from a distance it looks one-dimensional, but on closer inspection there is an additional circular direction. Describing the physics of the observable universe becomes a problem closely tied to the geometry of certain spaces, and conversely, demanding sensible physics as an output of string theory leads to new geometric techniques. One can then understand quantum corrections as coming from the string theory itself. String theory has led to new ideas in our understanding of QFT and gravity. We start with a strongly coupled QFT. Holography is an equivalence between two theories, let us call them theory A and theory B. Theory A is a d+1-dimensional gravity theory while theory B is QFT in flat (without gravity) d-dimensional space. Holography means that theory A can be utilised to learn about strong coupling aspects of theory B and vice versa. For example, theory B can be integrable, meaning it is completely soluble, and information about the strong coupling regime is determined purely by symmetry. Holography means we can determine (perhaps obscure) properties of theory A, the gravity theory. Conversely, via classical gravity, theory A can be used to describe regimes of strong coupling in theory B. Even if theory A nor theory B are not realistic models - one typically makes simplifying assumptions for the dualities to work -- one might hope the resulting features are universal, teaching us some new lessons on otherwise difficult problems in particle physics. Part I of this proposal is concerned with developing holography in a new paradigm of examples, as well as using integrability to explore properties of QFT. Next, in the context of gravity, new ideas have arisen in understanding black holes. Symmetries that derive from string theory, e.g. supersymmetry, have led to novel techniques for obtaining new types of black holes, as well as understanding their geometric and physical properties. Many interesting questions arise: what is the role of quantum corrections to these black hole solutions? Are they stable? A different but related question is how can we use string theory to describe quasi-realistic phenomenological models? Doing so requires understanding the geometry of spaces. What types of geometries lead to realistic models of our universe? What is the role of quantum corrections? These are the types of questions that form part II of this proposal.